`x=(1)/(3)+(1)/(3.3^(3))+(1)/(5.3^(5))+…..`
`y=(1)/(5)+(1)/(3.5^(3))+(1)/(5.5^(5))+….`
`z=1+(1)/(3.2^(2))+(1)/(5.2^(4))+(1)/(7.2^(6))+…..`
Then descending order of x, y, z

Options

A. a) z, y, x
B. b) z, x, y
C. c) x, y, z
D. d) y, z, x

Answer

To solve the problem, we need to evaluate the series \( x \), \( y \), and \( z \) and then compare their values to determine the descending order.

### Step 1: Evaluate \( x \)

The series for \( x \) is given by:
\[ x = \frac{1}{3} + \frac{1}{3 \cdot 3^3} + \frac{1}{5 \cdot 3^5} + \ldots \]

This can be expressed in terms of a logarithmic series. We can use the formula:
\[ \log\left(\frac{1+x}{1-x}\right) = 2x + \frac{2x^3}{3} + \frac{2x^5}{5} + \ldots \]

By substituting \( x = \frac{1}{3} \):
\[ x = \frac{1}{2} \log\left(\frac{1 + \frac{1}{3}}{1 - \frac{1}{3}}\right) = \frac{1}{2} \log\left(\frac{\frac{4}{3}}{\frac{2}{3}}\right) = \frac{1}{2} \log(2) \]

### Step 2: Evaluate \( y \)

The series for \( y \) is given by:
\[ y = \frac{1}{5} + \frac{1}{3 \cdot 5^3} + \frac{1}{5 \cdot 5^5} + \ldots \]

Using the same logarithmic series formula and substituting \( x = \frac{1}{5} \):
\[ y = \frac{1}{2} \log\left(\frac{1 + \frac{1}{5}}{1 - \frac{1}{5}}\right) = \frac{1}{2} \log\left(\frac{\frac{6}{5}}{\frac{4}{5}}\right) = \frac{1}{2} \log\left(\frac{3}{2}\right) \]

### Step 3: Evaluate \( z \)

The series for \( z \) is given by:
\[ z = 1 + \frac{1}{3 \cdot 2^2} + \frac{1}{5 \cdot 2^4} + \frac{1}{7 \cdot 2^6} + \ldots \]

Using the logarithmic series formula and substituting \( x = \frac{1}{2} \):
\[ z = \log\left(\frac{1 + \frac{1}{2}}{1 - \frac{1}{2}}\right) = \log\left(\frac{\frac{3}{2}}{\frac{1}{2}}\right) = \log(3) \]

### Step 4: Compare the values

Now we have:
- \( x = \frac{1}{2} \log(2) \)
- \( y = \frac{1}{2} \log\left(\frac{3}{2}\right) \)
- \( z = \log(3) \)

Next, we need to compare these values. We know that:
- \( \log(2) \approx 0.3010 \)
- \( \log\left(\frac{3}{2}\right) \approx 0.1761 \)
- \( \log(3) \approx 0.4771 \)

Calculating:
- \( x \approx \frac{1}{2} \cdot 0.3010 \approx 0.1505 \)
- \( y \approx \frac{1}{2} \cdot 0.1761 \approx 0.08805 \)
- \( z \approx 0.4771 \)

### Step 5: Determine the descending order

From the calculations:
- \( z \) is the largest,
- followed by \( x \),
- and then \( y \).

Thus, the descending order is:
\[ z > x > y \]